Definition 4 (Factor-width k matrices [1]}) 
The factor width of a symmetric matrix \(A\)  is the smallest integer \(k\)  such that there exists a matrix \(V\)  where \( A = VV^T\)  and each column of \( V\)  contains at most \(k\)  non-zeros. We denote the cone of \(n \times n\)  symmetric matrices of factor width \(k\)  by \( \mathcal {FW}_n^k\) .

